2022/09/09 by Zafer Şi̇ar, Şiar, Zafer, Refık Keskin +1
Physics and Astronomy · #Advanced Mathematical Theories and Applications #Black Holes and Theoretical Physics #FOS: Mathematics #Nonlinear Waves and Solitons #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2209.04190
openalex publication_date 2022/09/09 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Let k>=2 and let (Qn(k))n>=2-k be the k-generalized Pell sequence defined by Qn(k)=2Qn-1(k)+Qn-2(k)+...+Qn-k(k) for n>=2 with initial conditions Q-(k-2)(k)=Q-(k-3)(k)=...=Q-1(k)=0, Q0(k)=2,Q1(k)=2. In this paper, we solve the Diophantine equation Qn(k)=ym in positive integers n,m,y,k with m,y,k>=2. We show that all solutions (n,m,y) of this equation in positive integers n,m,y,k such that 2<=y<=100 are given by (n,m,y)=(3,2,4),(3,4,2) for k>=3. Namely, Q3(k)=16=24=42 for k>=3.